Question: How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \]
Solution: We take cases on the value of $x.$ If $x \le 1,$ then we have $(1-x) = (2-x) + (3-x),$ so $x = 4.$ But this does not satisfy $x<1,$ so it is not a valid solution.

If $1< x \le 2,$ then we have $x-1 = (2-x) + (3-x),$ so $x = 2,$ which is a valid solution.

If $2 < x \le 3,$ then we have $x-1 = (x-2) + (3-x),$ so $x=2$ again.

If $3 < x,$ then we have $(x-1) = (x-2) + (x-3),$ which gives $x=4.$ This time, $x=4$ is a valid solution because it satisfies $3 <x .$

Hence there are $\boxed{2}$ values of $x,$ namely $x=2$ and $x=4.$